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Interval table and short MOS discussion.
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It serves as a good first approximation to [[Nelinda#Xenharmonic Systems for Nelinda|nelindic temperament]], and is in many respects a "3n+1 cousin" of 5-limit [[12edo|12et]] (even though it takes every other step of the dissimilar [[27edo|27et]]), with relatively high error but low complexity, similar step size, and even sharing a common comma ([[128/125]]). Note the latter means that 27ed4 divides 4/1 into three approximate 8/5's, just as 12ed2 divides 2/1 into three 5/4's, and thus it has a 5/2 equally sharp of rational as the 5/4 in 12ed2. Its 7 and 13 approximations are a bit sharp themselves, and overall it lends itself well to IoE compression: the TE tuning gives one of 2395.819236 cents.
It serves as a good first approximation to [[Nelinda#Xenharmonic Systems for Nelinda|nelindic temperament]], and is in many respects a "3n+1 cousin" of 5-limit [[12edo|12et]] (even though it takes every other step of the dissimilar [[27edo|27et]]), with relatively high error but low complexity, similar step size, and even sharing a common comma ([[128/125]]). Note the latter means that 27ed4 divides 4/1 into three approximate 8/5's, just as 12ed2 divides 2/1 into three 5/4's, and thus it has a 5/2 equally sharp of rational as the 5/4 in 12ed2. Its 7 and 13 approximations are a bit sharp themselves, and overall it lends itself well to IoE compression: the TE tuning gives one of 2395.819236 cents.
== Intervals ==
The following table of intervals uses the 7-note (6L 1s) [[MOS scale]] of nelindic for the naturals, using a simple A-G notation and standard sharps/flats for the [[chroma]]. Extended to the 13-note (7L 6s) scale, these would include all of the sharps except for F#. Due to the L/s ratio of 3:1, in the 13-note case, most former diminished intervals become minor, most former minor intervals become augmented, and most former augmented intervals become major.
{| class="wikitable"
!'''Degree'''
!'''Note'''
!'''Interval name'''
!'''Cents'''
!'''~ Ratios'''
|-
|'''0'''
|'''A'''
|'''unison'''
|'''0'''
|'''1/1'''
|-
|1
|A#
|aug unison
|88.89
|21/20
|-
|2
|Bbb
|ddim mos2nd
|177.78
|10/9
|-
|3
|Bb
|dim mos2nd
|266.67
|7/6
|-
|''4''
|''B''
|''perf mos2nd''
|''355.56''
|''16/13''
|-
|5
|B#
|aug mos2nd
|444.44
|13/10, 9/7
|-
|6
|Cbb
|dim mos3rd
|533.33
|27/20, 19/14
|-
|7
|Cb
|min mos3rd
|622.22
|10/7, 13/9
|-
|8
|C
|maj mos3rd
|711.11
|3/2
|-
|9
|C#
|aug mos3rd
|800.00
|8/5
|-
|10
|Dbb
|dim mos4th
|888.89
|5/3
|-
|11
|Db
|min mos4th
|977.78
|7/4
|-
|12
|D
|maj mos4th
|1066.67
|13/7
|-
|13
|D#
|aug mos4th
|1155.56
|39/20, 35/18
|-
|14
|Ebb
|dim mos5th
|1244.44
|80/39, 72/35
|-
|15
|Eb
|min mos5th
|1333.33
|28/13
|-
|16
|E
|maj mos5th
|1422.22
|16/7
|-
|17
|E#
|aug mos5th
|1511.11
|12/5
|-
|18
|Fbb
|dim mos6th
|1600.00
|5/2
|-
|19
|Fb
|min mos6th
|1688.89
|8/3
|-
|20
|F
|maj mos6th
|1777.78
|14/5, 36/13
|-
|21
|F#
|aug mos6th
|1866.67
|80/27, 38/13
|-
|22
|Gb
|dim mos7th
|1955.56
|40/13, 28/9
|-
|''23''
|''G''
|''perf mos7th''
|''2044.44''
|''13/4''
|-
|24
|G#
|aug mos7th
|2133.33
|24/7
|-
|25
|Abb
|ddim tetratave
|2222.22
|18/5
|-
|26
|Ab
|dim tetratave
|2311.11
|80/21
|-
|'''27'''
|'''A'''
|'''tetratave'''
|'''2400'''
|'''4/1'''
|}
The [[Generator|genchain]] for the nelindic scale is as follows:
{| class="wikitable"
|Abb
|Bbb
|Cbb
|Dbb
|Ebb
|Fbb
|Gb
|Ab
|Bb
|Cb
|Db
|Eb
|Fb
|G
|A
|B
|C
|D
|E
|F
|G#
|A#
|B#
|C#
|D#
|E#
|F#
|-
|dd1
|dd2
|d3
|d4
|d5
|d6
|d7
|d1
|d2
|m3
|m4
|m5
|m6
|P7
|P1
|P2
|M3
|M4
|M5
|M6
|A7
|A1
|A2
|A3
|A4
|A5
|A6
|}
[[Category:Nonoctave]]
[[Category:Edonoi]]
[[Category:Ed4]]

Revision as of 16:44, 17 May 2021

27ed4 is an equal tuning that divides the 4/1 ratio (double-octave, tetratave, fifteenth) into steps of 88+(8/9) cents.

It serves as a good first approximation to nelindic temperament, and is in many respects a "3n+1 cousin" of 5-limit 12et (even though it takes every other step of the dissimilar 27et), with relatively high error but low complexity, similar step size, and even sharing a common comma (128/125). Note the latter means that 27ed4 divides 4/1 into three approximate 8/5's, just as 12ed2 divides 2/1 into three 5/4's, and thus it has a 5/2 equally sharp of rational as the 5/4 in 12ed2. Its 7 and 13 approximations are a bit sharp themselves, and overall it lends itself well to IoE compression: the TE tuning gives one of 2395.819236 cents.

Intervals

The following table of intervals uses the 7-note (6L 1s) MOS scale of nelindic for the naturals, using a simple A-G notation and standard sharps/flats for the chroma. Extended to the 13-note (7L 6s) scale, these would include all of the sharps except for F#. Due to the L/s ratio of 3:1, in the 13-note case, most former diminished intervals become minor, most former minor intervals become augmented, and most former augmented intervals become major.

Degree Note Interval name Cents ~ Ratios
0 A unison 0 1/1
1 A# aug unison 88.89 21/20
2 Bbb ddim mos2nd 177.78 10/9
3 Bb dim mos2nd 266.67 7/6
4 B perf mos2nd 355.56 16/13
5 B# aug mos2nd 444.44 13/10, 9/7
6 Cbb dim mos3rd 533.33 27/20, 19/14
7 Cb min mos3rd 622.22 10/7, 13/9
8 C maj mos3rd 711.11 3/2
9 C# aug mos3rd 800.00 8/5
10 Dbb dim mos4th 888.89 5/3
11 Db min mos4th 977.78 7/4
12 D maj mos4th 1066.67 13/7
13 D# aug mos4th 1155.56 39/20, 35/18
14 Ebb dim mos5th 1244.44 80/39, 72/35
15 Eb min mos5th 1333.33 28/13
16 E maj mos5th 1422.22 16/7
17 E# aug mos5th 1511.11 12/5
18 Fbb dim mos6th 1600.00 5/2
19 Fb min mos6th 1688.89 8/3
20 F maj mos6th 1777.78 14/5, 36/13
21 F# aug mos6th 1866.67 80/27, 38/13
22 Gb dim mos7th 1955.56 40/13, 28/9
23 G perf mos7th 2044.44 13/4
24 G# aug mos7th 2133.33 24/7
25 Abb ddim tetratave 2222.22 18/5
26 Ab dim tetratave 2311.11 80/21
27 A tetratave 2400 4/1

The genchain for the nelindic scale is as follows:

Abb Bbb Cbb Dbb Ebb Fbb Gb Ab Bb Cb Db Eb Fb G A B C D E F G# A# B# C# D# E# F#
dd1 dd2 d3 d4 d5 d6 d7 d1 d2 m3 m4 m5 m6 P7 P1 P2 M3 M4 M5 M6 A7 A1 A2 A3 A4 A5 A6
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